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  • mbenkerumass 6:00 am on April 3, 2020 Permalink | Reply

    Applications of the Paraxial Approximation 

    It was discussed in a previous article, Mirrors in Geometrical Optics, Paraxial Approximation that the paraxial approximation is used to consider an apparently imperfect or flawed system as a perfect system.

    Paraxial Approximation

    The paraxial approximation was proposed in response to a normal occurrence in optical systems where the focal point is inconsistent for incident rays of higher incidence angles.The focal point F for a spherical mirror is understood under the paraxial approximation to be half the radius of curvature. Without the paraxial approximation, the system becomes increasingly complicated, as the focal point is a varying trigonometric function of the angle of incidence. The paraxial approximation assumes that all incident angles will be small.


    The paraxial approximation can be likened (and when analyzed fully, this is it exactly) to a case of a triangle of base B, hypotenuse H and angle θ. Consider a case where H/B is very close to 1. θ will also be very small. In this case, it is of little harm to consider such a triangle as a triangle with θ=0, virtually to lines on top of each other, H and B, and more explicitly, H=B. This is precisely what is done when using the paraxial approximation.


    An interesting question to ask is, what angle should be the limit to which we allow a paraxial approximation? The answer would be, it depends on how accurate, or clear the image must be. When discussing optical systems, an aberration is a case in which rays are not precisely focused at the focal point of a mirror (or another type of optical system involving focusing). An aberration will actually cause the image clarity to be reduced at the output of the system. The following image would be an example of the result of an aberration to an image in an optical system:


    Here is an example of a problem that makes clear an example of the issue of an aberration. Two rays appear to be correctly aligned to the focal point, however another ray with angle of incidence of 55 degrees is not focused at the focal point. A system that would allow a ray of incidence of 55 degrees may be acceptable under some circumstances, however one would expect to have an aberration or some level of blurriness to the image.


  • mbenkerumass 5:00 am on March 28, 2020 Permalink | Reply

    Ray Tracing Examples (1) Curved Mirrors 

    The following ray tracing examples all utilize Fermat’s principle in examining ray traces incident at a mirror.

    Example 1. Draw a ray trace for a ray angled at a convex mirror.

    The ray makes a 40 degree angle with the normal of the mirror at the point of incidence. In accordance with the law of reflection (Fermat’s Principle), the ray will exit at 40 degrees on the other side of the normal.



    The above example shows a single ray at an angle. Often, rays are drawn together in a group of parrallel rays. This example shows how an incident set of parallel rays will no longer be parallel when reflected by a non-uniform (not flat) mirror surface.



    This example brings up an important concept that happens especially with concave mirrors. Two rays drawn seem to be directed towards the same point, known as the focal point. A focal point however is only consistent for smaller angles. The third ray at the bottom makes a 55 degree incident angle with the normal of the surface. The reflected ray is also 55 degrees separated from the normal but is directed to the other side of the normal. The ray does not converge at the focal point as the others do. This effect is known as an aberration and may be discussed further at length in a later article.



    This example makes use of the above concept of focal point. An object placed at the focal point will not make an image at the focal point. This is useful if for instance, some type of lense or collecter should be placed at the focus of the mirror. This can be done without worry for it causing disturbances to the image that is formed at the focal point by the reflected rays.


  • mbenkerumass 5:01 am on March 27, 2020 Permalink | Reply

    Principles of Ray Tracing (1) 

    In geometrical optics, light is treated as rays, typically drawn as lines that propagate in a straight line from one point to another. Ray tracing is a method of determining how a ray will react to a surface or mirror. Rays are understood to propagate always in a straight line, however when entering an angled surface, rebounding from an angled surface or propagating through a different medium, there are a few techniques that are needed to reliably determine the direction and path of a light ray. The following properties are the basis for ray tracing.

    Refractive Index

    The refractive index is a property intrinsic to a medium that describes how fast or slow light propagates in the medium. Light speed in a vacuum is 3*10^8 m/s. Light speed will only get slower in real mediums. The formula for refractive index is the speed of light c devided by the velocity of light in the medium.


    The refractive index of air is approximately 1. The refractive index of glass for instance is about 1.5. This has implications on how light will propage when changing from one medium to another.




    Snell’s Law

    Snell’s law uses the angle of incidence (incoming ray), the angle of refraction (exiting ray) and the refractive indexes of each medium at a boundary to determine the path of propagation. Consider the example below:


    Snell’s Law: η1*sin(θ1) = η2*sin(θ2)

    The angle of incidence and the angle of refraction are both with respect to the normal of the surface!


    Fermat’s Principle

    Fermat’s Principle is also demonstrated in the above figure. Fermat’s Principle states that the angle of incidence of a ray will be equal to the angle of reflection, but exiting from the other side of the normal of the surface.


    Using these principles alone, many optical instruments and technologies can be designed and built that manipulate the direction of light rays.

  • mbenkerumass 5:00 am on March 18, 2020 Permalink | Reply

    Hermitian Operators, Time-Shifting Wavefunction 

    It was mentioned in the previous article on Quantum Mechanics [link] that if the integral of a wavefunction over all space at one time is equal to one (thereby meaning that it is normalized and that the probabilility of the particle existing is 100%), then the wavefunction is applicable to a later time, t.


    A function in the place of Ψ*Ψ is used as a probability density function, ρ(x,t). The function N(t) is the resultant probability at a given time, given that the probability was found to be equal to 1 at a given time t0. Shown below, it is proposed that for dN/dt to equal zero, the Hamiltonian must be a Hermitian operator.


    A Hermitian operator would satisfy the following:


    Hermiticity in general may referred to as a type of conjugate form of an operator. An operator is hermitian if the hermitian conjugate is equal to itself. One may compare this relationship as to a real number whose complex conjugate is equal to itself.


    Returning to the calculation of dN/dt,


  • mbenkerumass 5:00 am on March 17, 2020 Permalink | Reply

    Ψ Wavefunction Describes Probability 

    Schroedinger’s first interpretation of the wavefunction was that Ψ would describe how a particle dissipates. Where the wavefunction Ψ was the highest, then that was where more of the particle was present. Max Born disagreed saying that a particle would not dissintegrates, choosing another direction to move. Max Born proposed that the wavefunction would actually describe the probability of a particle inhabiting a space. Both Schroedinger and Einstein were initially opposed to the idea of a probabilistic interpretation of the Schroedinger equation. The probabilistic interpretation of Max Born however later became the consensus view of quantum mechanics.

    The wavefunction Ψ therefore describes the probability of finding a particle at position x at time t, not the amount of the particle that exists there.


    Since the Schroedinger equation is both a function of position and time, it can only be solved for one variable at a time. Solving for position is preferable due to the fact that if the wavefunction is known for all x, this can provide information for how the wavefunction is at a later time.


    Of the limits regarding the wavefunction, it is also said that the wavefunction must be convergent. The wavefunction therefore does not approach a finite constant as x approaches infinity.


    We also recall that a wavefunction may also be multiplied by a number. It would appear that doing so would violate the above expression. The answer regarding this conjecture is that the above formula represents a normalized wavefunction. Yet it turns out that not all wavefunctions are normalizable. The case of multiplying the wavefunction with a magnitude in fact would still be normalizable, however. A wavefunction can be normalized if the integral is a finite number less than infinity using the following method:




  • mbenkerumass 5:00 am on March 16, 2020 Permalink | Reply

    Matrices, Multiple Dimensions in Quantum Mechanics 

    There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.


    Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:


    This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.


    Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.



  • mbenkerumass 5:00 am on March 15, 2020 Permalink | Reply

    Operators in Quantum Mechanics 

    Before getting into problems relating to the free particle schroedinger equation, let’s review the full Schroedinger equation. The energy operator E_hat appears in the first equation below. Thus far, the euqation relates only kinetic energy. Potential energy however when considered would allow the Schroedinger equation to be applied in a wide range of possible applications, being able to describe the interactions of atoms and molecules and their interactions in free space, wells, and other environments due to the linearity of quantum mechanics. One major point to take from discovering the free particle Schroedinger equation is how important it is in Quantum Mechanics to create energy operators. An operator can be as simple as a constant or as complicated as a partial differential. By allowing an ‘operator’ to take on this wider range of features as opposed to a basic variable makes for the basis of many quantum mechanical calculations. It then follows that the portential, V(x,t) can also be treated as an operator that modifies the system.


    Consider an operator X_hat that when multiplied by a function, results in the function being multiplied by x. Remember that although this may look like a variable, it is useful to consider this as an operator in Quantum mechanics.


    Does the order in which operators are multiplied matter?


    Considering that operators are not always constants or variables, but also sometimes differentials, the order of operations for operators does matter.


    A communtator is understood as the difference of linear operators. The commutator of x_hat and p_hat is i*h_bar.


  • mbenkerumass 5:00 am on March 14, 2020 Permalink | Reply
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    Direct-Bandgap & Indirect-Bandgap Semiconductors 

    Direct Semiconductors

    When light reaches a semiconductor, the light is absorbed if the photon energy is greater than or equal to the band gap, creating electron-hole pairs. In a direct semiconductor, the minimum of the conduction band is aligned with the maximum of the valence band.



    One example of a direct semiconductor is GaAs. The band diagram for GaAs is shown to

    the right. As the gap between the valence band and conduction band is 1.42eV, if a

    photon of same or greater energy is applied to the semiconductor, a hole-electron pair is created for each photon. This is termed the photo-excitation of semiconductors. The photon is thereby absorbed into the semiconductor.




    Indirect Semiconductors and Phonons

    indiresemicFor an indirect semiconductor to absorb a photon, the process must be mediated by phonons, which are quanta of sound and in this case refer to the acoustic vibration of crystal lattice. A phonon is also used to provide energy for radiative recombination. When understanding the essence of a phonon, one should recall that sound is not necessarily within hearing range (20 – 20kHz). In fact, the sound vibrations in a semiconductor may well be in the Terrahertz range. The diagram to the right shows how an indirect semiconductor band would appear and also the use of phonon energy to mediate the process of allowing the indirect semiconductor to behave as a semiconductor.



    Excitons are bound electron-hole pairs that are created in pure semiconductors when a photon with bandgap energy or larger is absorbed. In bulk semiconductors, these excitons will dissipate rapidly. In quantum wells however, the excitons may remain, even at room temperature. The effect of the quantum well is to force an electron and hole to be very close to each other. This allows for a strong bonding effect to take place and allows the quantum well the ability to generate light as a semiconductor laser.



    The band structure of a semiconductor is given by:


    Where mc = 0.2 * m0 and mv = 0.8 * m0 and Eg = 1.6 eV. Sketch the E-k Diagram.


  • mbenkerumass 5:00 am on March 13, 2020 Permalink | Reply

    DeBroglie Relations and the Scale of Quantum Effects (MIT OpenCourseWare) 

    Assignment Sheet MIT OpenCourseWare – Quantum Physics I



    PDF of solutions

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.


  • mbenkerumass 5:00 am on March 11, 2020 Permalink | Reply

    Ray Tracing with Snell’s Law – Optics, ECE591 

    Of the four ways of manipulating light, these examples employ shaping of a lens and the refractive index to change the path of a ray.




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